Holomorphic structures on the quantum projective line

Abstract

We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted positivity for twisted Hochschild and cyclic cocycles and exhibit an explicit twisted positive Hochschild cocycle for the complex structure on the sphere.

22 pages, no figures. Published in IMRN

Cited by in corpus (16)

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• The homogeneous coordinate ring of the quantum projective plane
• Kähler structures on quantum irreducible flag manifolds
• Noncommutative homogeneous spaces: the matrix case
• Calculi, Hodge operators and Laplacians on a quantum Hopf fibration
• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Noncommutative Differential Geometry of Generalized Weyl Algebras
• Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds
• A Kodaira Vanishing Theorem for Noncommutative Kahler Structures
• Kähler structure on certain $C^{\ast }$-dynamical systems and the noncommutative even dimensional tori
• Twisted sigma-model solitons on the quantum projective line
• A twisted local index formula for curved noncommutative two tori
• Almost complex structure on finite points from bidirected graphs
• Fubini-Study metrics and Levi-Civita connections on quantum projective spaces
• Differential and holomorphic differential operators on noncommutative algebras
• Noncommutative Kahler Geometry of the Standard Podles Sphere